Study of the Numerical Method for an Inverse Problem of a Simplified Intestinal Crypt

In this chapter we consider the study of an inverse problem for an intestinal crypt model. The original model is based on the interaction of epithelial cells with microbiota-derived chemicals diffusing in the crypt from the gut lumen. The five types of cells considered in the original model were reduced in this work to three types of cells for simplifications of the inverse problem. The inverse problem consists of determining the shape of the secretory cells of the deep crypt from observations of the stem cells and progenitor cells at a fixed time. The method used is the calculation of the adjoint state associated with the second-order BGK numerical scheme considered in Darrigade et al. (J Math Biol 84(7):60;2022), which allows calculating the critical points of the Lagrangian associated with the inverse problem and applying a gradient method in order to minimize the cost function. The algorithm is described, and some numerical examples are given.